Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

Dynamic Equations on Time Scales: An Introduction with Applications

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Dedication We dedicate this book to our family members who complete us. In particular, M. Ben-chohra's dedication is to his wife, Kheira, and his children, Mohamed, Maroua, and Abdelillah; J. Henderson dedicates to his wife, Darlene, and his descendants, Kathy, Contents Preface xi 1. Preliminaries 1 1.1. Definitions and results for multivalued analysis 1 1.2. Fixed point theorems 4 1.3. Semigroups 7 1.4. Some additional lemmas and notions 9 2. Impulsive ordinary differential equations & inclusions 11 2.1. Introduction 11 2.2. Impulsive ordinary differential equations 12 2.3. Impulsive ordinary differential inclusions 24 2.4. Ordinary damped differential inclusions 49 2.5. Notes and remarks 62 3. Impulsive functional differential equations & inclusions 63 3.1. Introduction 63 3.2. Impulsive functional differential equations 63 3.3. Impulsive neutral differential equations 74 3.4. Impulsive functional differential inclusions 80 3.5. Impulsive neutral functional DIs 95 3.6. Impulsive semilinear functional DIs 107 3.7. Notes and remarks 118 4. Impulsive differential inclusions with nonlocal conditions 119 4.1. Introduction 119 4.2. Nonlocal impulsive semilinear differential inclusions 119 4.3. Existence results for impulsive functional semilinear differential inclusions with nonlocal conditions 136 4.4. Notes and remarks 145 5. Positive solutions for impulsive differential equations 147 5.1. Introduction 147 5.2. Positive solutions for impulsive functional differential equations 147 5.3. Positive solutions for impulsive boundary value problems 154 5.4. Double positive solutions for impulsive boundary value problems 159 5.5. Notes and remarks 165 viii Contents 6. Boundary value problems for impulsive differential inclusions 167 6.1. Introduction 167 6.2. First-order impulsive differential inclusions with periodic boundary conditions 167 6.3. Upper-and lower-solutions method for impulsive differential inclusions with nonlinear boundary conditions 184 6.4. Second-order boundary value problems 191 6.5. Notes and remarks 198 7. Nonresonance impulsive differential inclusions 199 7.1. Introduction 199 7.2. Nonresonance first-order impulsive functional differential inclusions with periodic boundary conditions 199 7.3. Nonresonance second-order impulsive functional differential inclusions with periodic boundary conditions 209 7.4. Nonresonance higher-order boundary value problems for impulsive functional differential inclusions 217 7.5. Notes and remarks 227 8. Impulsive differential equations & inclusions with variable times 229 8.1. Introduction 229 8.2. First-order impulsive differential equations with variable times 229 8.3. Higher-order impulsive differential equations with variable times 235 8.4. Boundary value problems for differential inclusions with variable times 241 8.5. Notes and remarks 252 9. Nondensely defined impulsive differential equations & inclusions 253 9.1. Introduction 253 9.2. Nondensely defined impulsive semilinear differential equations with nonlocal conditions 253 9.3. Nondensely defined …

Impulsive Differential Equations and Inclusions

高等学校計算数学学報 = Numerical mathematics

https://web.mst.edu/~bohner/papers/deotsas.pdf

Dynamic equations on time scales: a survey

A higher order numerical scheme for generalized fractional diffusion equations

We consider the following boundary value problem¶¶$ \left\{ \begin{gathered} \Delta ^n u_i (k) + q_i (k,u_1 (k),u_2 (k), \cdots ,u_m (k)) = 0,k \in \{ 0,1, \cdots ,N\} \hfill \\ \Delta ^j u_i (0) = 0 = \Delta ^{p_i } u_i (N + n - p_i ),0 \leqslant j \leqslant n - 2 \hfill \\ i = 1,2, \cdots ,m \hfill \\ \end{gathered} \right. $¶where $ n\geq 2,~N\geq n-1,~1\leq p_i\leq n-1,~1\leq i\leq m $, and the nonlinearities $ q_i,~1\leq i\leq m $, can be 'nonpositive'. Several criteria are offered for the existence of single and double solutions of the system that are 'positive'.

Positive solutions for a system of nonpositive difference equations

We consider the following system of higher order three-point boundary-value problems u"i^(^m^"^i^)(t)=f"i(t,u"1(@f"1(t)),u"2(@f"2(t)),...,u"n(@f"n(t))),[email protected]?[a,b]u"i^(^j^)(a)=0,[email protected][email protected]?m"i-3u"i^(^m^"^i^-^2^)(t^*)=0,@xu"i^(^m^"^i^-^3^)(b)[email protected]"i^(^m^"^i^-^1^)(b)=0 where i=1,2,...,n, m"i>=3, 12(a+b)=0, @d>0 and @f"i's are deviating arguments. Several criteria are offered for the existence of three fixed-sign solutions (u"1,u"2,...,u"n) of the system, i.e. for each [email protected][email protected]?n and all [email protected]?[a,b], @q"iu"i(t)>=0 where @q"[email protected]?{1,-1} is predetermined. We also present some examples to illustrate the results obtained.

Multiple fixed-sign solutions for a system of higher order three-point boundary-value problems with deviating arguments

We consider the system of difference equations 

$$u_i (k) = \lambda \mathop \sum \limits_{\ell = 0}^N g_i (k,\ell )P_i (\ell ,u_1 (\ell ),u_2 (\ell ),...,u_n (\ell )), k \in \{ 0,1,...,T\} , 1 \leqslant i \leqslant n,$$

where λ > 0 and T ≥ N ≥ 0. Our aim is to determine the values of λ for which the above system has a constant-sign solution. In addition, explicit intervals for λ are presented. The generality of the results obtained is illustrated through applications to several well-known boundary-value problems. We also extend the above problem to that on {0, 1, ...}: 

$$u_i (k) = \lambda \mathop \sum \limits_{\ell = 0}^\infty g_i (k,\ell )P_i (\ell ,u_1 (\ell ),u_2 (\ell ),...,u_n (\ell )), k \in \{ 0,1,...,T\} , 1 \leqslant i \leqslant n.$$

Finally, both systems above are extended to the general case where λ is replaced by λ i .

/pdf/eigenvalue-characterization-of-a-system-of-difference-2lhy7re3yc.pdf

Patricia J. Y. Wong

Papers

A higher order numerical scheme for generalized fractional diffusion equations

Positive solutions for a system of nonpositive difference equations

Multiple fixed-sign solutions for a system of higher order three-point boundary-value problems with deviating arguments

Eigenvalue Characterization of a System of Difference Equations

Comparison theorems for the oscillation of higher order difference equations with deviating arguments